Problem 11E

Interval and radius of convergence Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence.

Solution 11E

Step 1:

Given series is and

To determine the radius of convergence we use ratio test which states:

The ratio test states that:

a. If then the series converges

b. If then the series diverges

c. If or the limit does not exist then the test is inconclusive.

Calculating L, we get

Since and independent of x, we can conclude that the series converges for all x.

Thus the interval of convergence is the interval . The radius of convergence in this case is said to be .